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r36/xlog/CRACK.LOG
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2011-09-02 18:13:33
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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... load_package crackapp; % Needed for these tests. % Initial load up list off echo$ .......................................................................... An example of the determination of point symmetries for ODEs -------------------------------------------------------------------------- This is LIEPDE - a program for calculating infinitesimal symmetries of single ODEs/PDEs and ODE/PDE - systems The ODE/PDE (-system) under investigation is : 4 3 2 0 = df(y,x,2)*x - df(y,x)*x - 2*df(y,x)*x*y + 4*y for the function(s) : y(x) time to formulate conditions: 190 ms GC time : 0 ms This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: expr. with 21 terms functions: eta_y(y,x) xi_x(y,x) separation w.r.t. y|1 new function: c1(x) new function: c2(x) integrated equation : 0=c1 + c2*y + xi_x separation yields 4 equations substitution : xi_x= - c1 - c2*y eta_y(y,x) xi_x= - c1 - c2*y new function: c3(x) new function: c4(x) new function: c5(x) new function: c6(x) integrated equation : 3 2 2 2 3 3 0=3*df(c2,x)*x *y + 3*c2*x *y + 2*c2*y + 3*c5 + 3*c6*y + 3*eta_y*x substitution : 3 2 2 2 3 - 3*df(c2,x)*x *y - 3*c2*x *y - 2*c2*y - 3*c5 - 3*c6*y eta_y=------------------------------------------------------------ 3 3*x 3 2 2 2 3 - 3*df(c2,x)*x *y - 3*c2*x *y - 2*c2*y - 3*c5 - 3*c6*y eta_y=------------------------------------------------------------ 3 3*x xi_x= - c1 - c2*y separation w.r.t. y separation yields 4 equations substitution : c2=0 - c5 - c6*y eta_y=-------------- 3 x xi_x= - c1 substitution : 3 2 c6= - df(c1,x)*x + 3*c1*x 3 2 df(c1,x)*x *y - 3*c1*x *y - c5 eta_y=-------------------------------- 3 x xi_x= - c1 substitution : 6 5 4 - 3*df(c1,x,2)*x + 5*df(c1,x)*x - 5*c1*x c5=---------------------------------------------- 2 4 3 2 3*df(c1,x,2)*x - 5*df(c1,x)*x + 2*df(c1,x)*x*y + 5*c1*x - 6*c1*y eta_y=--------------------------------------------------------------------- 2*x xi_x= - c1 separation w.r.t. y new constant: c7 new constant: c8 integrated equation : 0=log(x)*c8*x - c1 + c7*x new constant: c9 new constant: c10 new constant: c11 integrated equation : 3 0=log(x)*c10*x - c1 + c11*x + c9*x new constant: c12 new constant: c13 new constant: c14 new constant: c15 integrated equation : 2/3 2 2 2 0=x *c14*x + log(x)*c13*x - c1*x + c12*x + c15 separation yields 3 equations substitution : c1=log(x)*c8*x + c7*x 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x separation w.r.t. x linear independent expressions : x*log(x) 3 x x separation yields 3 equations substitution : c11=0 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x separation w.r.t. x linear independent expressions : 2 2/3 x *x 2 x *log(x) 2 x 1 separation yields 4 equations substitution : c14=0 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x substitution : c15=0 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x substitution : c12=c7 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x substitution : c13=c8 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x substitution : c10=c8 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x substitution : c9=c7 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y xi_x= - log(x)*c8*x - c7*x End of this CRACK run The solution : xi_x= - log(x)*c8*x - c7*x 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y Free functions or constants : c7 c8 ************************************************************************** CRACK needed : 2080 ms GC time : 250 ms Remaining free functions after the last CRACK-run: c7 c8 The symmetries are: xi_x= - log(x)*c8*x - c7*x 2 eta_y= - 2*log(x)*c8*y - 2*c7*y - c8*x + c8*y with c7 c8 which are free. .......................................................................... An example of the determination of point symmetries for PDEs -------------------------------------------------------------------------- This is LIEPDE - a program for calculating infinitesimal symmetries of single ODEs/PDEs and ODE/PDE - systems The ODE/PDE (-system) under investigation is : 0 = df(u,x,2) - df(u,y) for the function(s) : u(y,x) time to formulate conditions: 170 ms GC time : 0 ms This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: 0= - 2*df(xi_y,u)*u|1 - 2*df(xi_y,x) functions: eta_u(u,y,x) xi_y(u,y,x) xi_x(u,y,x) separation w.r.t. u|1 new function: c18(u,y) new function: c19(u,y) integrated equation : 0=c19 + xi_y new function: c20(y,x) new function: c21(y,x) integrated equation : 0=c21 + xi_y separation yields 2 equations substitution : xi_y= - c21 eta_u(u,y,x) xi_y= - c21 xi_x(u,y,x) generalized separation new function: c22(y) new function: c23(y) separation yields 3 equations substitution : c21=c22 eta_u(u,y,x) xi_y= - c22 xi_x(u,y,x) substitution : c19=c22 eta_u(u,y,x) xi_y= - c22 xi_x(u,y,x) substitution : c22= - c23 eta_u(u,y,x) xi_y=c23 xi_x(u,y,x) End of this CRACK run The solution : xi_y=c23 Free functions or constants : xi_x(u,y,x) eta_u(u,y,x) c23(y) ************************************************************************** CRACK needed : 120 ms GC time : 0 ms Remaining free functions after the last CRACK-run: xi_x(u,y,x) eta_u(u,y,x) c23(y) time to formulate conditions: 100 ms GC time : 0 ms This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: expr. with 12 terms functions: xi_x(u,y,x) eta_u(u,y,x) c23(y) separation w.r.t. u|1 separation w.r.t. u|2 separation w.r.t. u|2 new function: c24(y,x) new function: c25(y,x) new function: c26(y,x) new function: c27(y,x) integrated equation : 0=c26 + c27*u + xi_x new function: c28(y,x) integrated equation : 0=df(eta_u,u) - 2*df(xi_x,x) + c28 new function: c29(y,x) new function: c30(y,x) integrated equation : 0=c30 + xi_x new function: c31(u,y) new function: c32(u,y) integrated equation : 0=df(c23,y)*x + c32 - 2*xi_x separation yields 6 equations substitution : xi_x= - c30 xi_x= - c30 eta_u(u,y,x) c23(y) generalized separation new function: c33(y) new function: c34(y) separation yields 3 equations substitution : c32= - c34 xi_x= - c30 eta_u(u,y,x) c23(y) substitution : - df(c23,y)*x + c34 c30=---------------------- 2 df(c23,y)*x - c34 xi_x=------------------- 2 eta_u(u,y,x) c23(y) substitution : c33= - c34 df(c23,y)*x - c34 xi_x=------------------- 2 eta_u(u,y,x) c23(y) new function: c35(x,y) new function: c36(x,y) integrated equation : 0=df(c23,y)*u - c28*u + c36 - eta_u substitution : eta_u=df(c23,y)*u - c28*u + c36 df(c23,y)*x - c34 xi_x=------------------- 2 eta_u=df(c23,y)*u - c28*u + c36 c23(y) new function: c37(y) new function: c38(y) integrated equation : 2 0=df(c23,y,2)*x - 2*df(c34,y)*x - 8*c28 + 2*c38 substitution : 2 df(c23,y,2)*x - 2*df(c34,y)*x + 2*c38 c28=---------------------------------------- 8 df(c23,y)*x - c34 xi_x=------------------- 2 2 - df(c23,y,2)*u*x + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u eta_u=------------------------------------------------------------------------- 8 c23(y) separation w.r.t. u separation yields 2 equations substitution : c27=0 df(c23,y)*x - c34 xi_x=------------------- 2 2 - df(c23,y,2)*u*x + 8*df(c23,y)*u + 2*df(c34,y)*u*x + 8*c36 - 2*c38*u eta_u=------------------------------------------------------------------------- 8 c23(y) separation w.r.t. u separation w.r.t. x new constant: c39 new constant: c40 new constant: c41 new constant: c42 new constant: c43 new constant: c44 integrated equation : 2 0=2*c23 + 2*c42 + c43*y + 2*c44*y new constant: c45 new constant: c46 new constant: c47 new constant: c48 integrated equation : 0=c34 + c47 + c48*y new constant: c49 integrated equation : 0=5*df(c23,y) - c38 + c49 separation yields 4 equations substitution : c34= - c47 - c48*y df(c23,y)*x + c47 + c48*y xi_x=--------------------------- 2 2 - df(c23,y,2)*u*x + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x eta_u=------------------------------------------------------------------- 8 c23(y) substitution : - df(c23,y)*x - c47 - c48*y c26=------------------------------ 2 df(c23,y)*x + c47 + c48*y xi_x=--------------------------- 2 2 - df(c23,y,2)*u*x + 8*df(c23,y)*u + 8*c36 - 2*c38*u - 2*c48*u*x eta_u=------------------------------------------------------------------- 8 c23(y) substitution : 2 - 2*c42 - c43*y - 2*c44*y c23=----------------------------- 2 - c43*x*y - c44*x + c47 + c48*y xi_x=---------------------------------- 2 2 8*c36 - 2*c38*u + c43*u*x - 8*c43*u*y - 8*c44*u - 2*c48*u*x eta_u=-------------------------------------------------------------- 8 2 - 2*c42 - c43*y - 2*c44*y c23=----------------------------- 2 substitution : c38= - 5*c43*y - 5*c44 + c49 - c43*x*y - c44*x + c47 + c48*y xi_x=---------------------------------- 2 2 8*c36 + c43*u*x + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u eta_u=-------------------------------------------------------------- 8 2 - 2*c42 - c43*y - 2*c44*y c23=----------------------------- 2 decoupling: c36 new equations: End of this CRACK run The solution : 2 - 2*c42 - c43*y - 2*c44*y c23=----------------------------- 2 2 8*c36 + c43*u*x + 2*c43*u*y + 2*c44*u - 2*c48*u*x - 2*c49*u eta_u=-------------------------------------------------------------- 8 - c43*x*y - c44*x + c47 + c48*y xi_x=---------------------------------- 2 Remaining conditions : 0=df(c36,x,2) - df(c36,y) for the functions : c42 c47 c48 c43 c44 c49 c36(x,y) ************************************************************************** CRACK needed : 980 ms GC time : 110 ms Remaining free functions after the last CRACK-run: c42 c47 c48 c43 c44 c49 c36(x,y) Free constants and/or functions have been rescaled. The symmetries are: xi_x= - 4*c43*x*y - 2*c44*x + c47 + 2*c48*y 2 xi_y= - c42 - 4*c43*y - 4*c44*y 2 eta_u=c36 + c43*u*x + 2*c43*u*y + c44*u - c48*u*x - c49*u with c42 c47 c48 c43 c44 c49 c36(x,y) which still have to satisfy: 0=df(c36,x,2) - df(c36,y) .......................................................................... An example of the determination of first integrals of ODEs Determination of a first integral for: 2 2 2 df(y,x) *x - 2*df(y,x) - y df(y,x,2)=------------------------------ x new function: h_0(y,x) new function: h_1(y,x) new function: h_2(y,x) 2 of the type: df(y,x) *h_2 + df(y,x)*h_1 + h_0 This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: expr. with 13 terms functions: h_2(y,x) h_1(y,x) h_0(y,x) separation w.r.t. d_y(1) new function: c50(x) integrated equation : 2*x*y 0=e *h_2 - c50 separation yields 4 equations substitution : c50 h_2=-------- 2*x*y e c50 h_2=-------- 2*x*y e h_1(y,x) h_0(y,x) substitution : df(h_0,x)*x h_1=------------- 2 y c50 h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) new function: c51(x) integrated equation : 2 2*x*y 3 x*y 3 2 3 2 0=df(c50,x)*x*y - e *df(h_0,x)*x + e *c51*x *y - 2*c50*x*y - 6*c50*y decoupling: h_0 new equations: expr. with 10 terms with leading derivative (df h_0 x 3) replaces a de with (df h_0 y) expr. with 20 terms with leading derivative (df h_0 x 2) replaces a de with (df h_0 x 3) expr. with 17 terms with leading derivative (df h_0 x) replaces a de with (df h_0 x 2) equations: expr. with 13 terms 2*x*y 2 2*x*y 2*x*y 2 4 0=e *df(h_0,x,2)*x - e *df(h_0,x)*x + e *df(h_0,y)*x*y - 2*c50*y 2 2*x*y 3 x*y 3 2 3 2 0=df(c50,x)*x*y - e *df(h_0,x)*x + e *c51*x *y - 2*c50*x*y - 6*c50*y functions: c50 h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) c51(x) c50(x) separation w.r.t. y linear independent expressions : 1 y 2 y x*y e x*y y*e new constant: c52 integrated equation : 0=c51 - c52*x new constant: c53 new constant: c54 integrated equation : 0=c51 + c53 + c54*x new constant: c55 integrated equation : 4 0=c50 - c55*x new constant: c56 new constant: c57 integrated equation : 4 4 0=log(x)*c57*x - c50 + c56*x new constant: c58 new constant: c59 new constant: c60 integrated equation : 6 4 3 0=c50 - c58*x - c59*x - c60*x separation yields 5 equations substitution : c51=c52*x c50 h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) separation w.r.t. x separation yields 2 equations substitution : c53=0 c50 h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) substitution : 4 c50=c55*x 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) separation w.r.t. x linear independent expressions : 4 x *log(x) 4 x separation yields 2 equations substitution : c57=0 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) new function: c61(y) substitution : c56=c55 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) substitution : c54= - c52 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) separation w.r.t. x separation yields 3 equations substitution : c60=0 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) substitution : c58=0 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) substitution : c59=c55 4 c55*x h_2=-------- 2*x*y e df(h_0,x)*x h_1=------------- 2 y h_0(y,x) decoupling: h_0 new equations: new function: c62(y) new function: c63(y) new function: c64(y) new function: c65(y) integrated equation : expr. with 10 terms substitution : 2*sqrt(2)*x*y + 2*x*y sqrt(2)*x*y + 2*x*y h_0=(e *sqrt(2)*c62 - 2*e *c65*y sqrt(2)*x*y + x*y 2 sqrt(2)*x*y + x*y - 2*e *c52*x*y - 2*e *c52*y sqrt(2)*x*y 2 3 sqrt(2)*x*y 2 + 2*e *c55*x *y + 4*e *c55*x*y sqrt(2)*x*y 2*x*y sqrt(2)*x*y + 2*x*y + 2*e *c55*y - e *sqrt(2)*c63)/(2*e *y) 4 c55*x h_2=-------- 2*x*y e 2*sqrt(2)*x*y + 2*x*y sqrt(2)*x*y + x*y 2 2 h_1=(e *c62*x + e *c52*x *y sqrt(2)*x*y 3 3 sqrt(2)*x*y 2 2 2*x*y - 2*e *c55*x *y - 2*e *c55*x *y + e *c63*x)/( sqrt(2)*x*y + 2*x*y 2 e *y ) h_0= expr. with 9 terms equations: 2*sqrt(2)*x*y + 2*x*y 2*x*y e *c62 + e *c63 0=----------------------------------------- sqrt(2)*x*y e expr. with 12 terms functions: 4 c55*x h_2=-------- 2*x*y e 2*sqrt(2)*x*y + 2*x*y sqrt(2)*x*y + x*y 2 2 h_1=(e *c62*x + e *c52*x *y sqrt(2)*x*y 3 3 sqrt(2)*x*y 2 2 2*x*y - 2*e *c55*x *y - 2*e *c55*x *y + e *c63*x)/( sqrt(2)*x*y + 2*x*y 2 e *y ) h_0= expr. with 9 terms c62(y) c63(y) c65(y) separation w.r.t. x linear independent expressions : 2*x*y + 2*sqrt(2)*x*y e 2*x*y e separation yields 2 equations substitution : c62=0 4 c55*x h_2=-------- 2*x*y e sqrt(2)*x*y + x*y 2 2 sqrt(2)*x*y 3 3 h_1=(e *c52*x *y - 2*e *c55*x *y sqrt(2)*x*y 2 2 2*x*y sqrt(2)*x*y + 2*x*y 2 - 2*e *c55*x *y + e *c63*x)/(e *y ) sqrt(2)*x*y + 2*x*y sqrt(2)*x*y + x*y 2 h_0=( - 2*e *c65*y - 2*e *c52*x*y sqrt(2)*x*y + x*y sqrt(2)*x*y 2 3 - 2*e *c52*y + 2*e *c55*x *y sqrt(2)*x*y 2 sqrt(2)*x*y 2*x*y + 4*e *c55*x*y + 2*e *c55*y - e *sqrt(2)*c63)/(2 sqrt(2)*x*y + 2*x*y *e *y) substitution : c63=0 4 c55*x h_2=-------- 2*x*y e x*y 2 3 2 e *c52*x - 2*c55*x *y - 2*c55*x h_1=------------------------------------- 2*x*y e 2*x*y x*y x*y 2 2 - e *c65 - e *c52*x*y - e *c52 + c55*x *y + 2*c55*x*y + c55 h_0=----------------------------------------------------------------------- 2*x*y e new constant: c66 integrated equation : 0=c65 + c66 substitution : c65= - c66 4 c55*x h_2=-------- 2*x*y e x*y 2 3 2 e *c52*x - 2*c55*x *y - 2*c55*x h_1=------------------------------------- 2*x*y e 2*x*y x*y x*y 2 2 e *c66 - e *c52*x*y - e *c52 + c55*x *y + 2*c55*x*y + c55 h_0=-------------------------------------------------------------------- 2*x*y e End of this CRACK run The solution : 2*x*y x*y x*y 2 2 e *c66 - e *c52*x*y - e *c52 + c55*x *y + 2*c55*x*y + c55 h_0=-------------------------------------------------------------------- 2*x*y e x*y 2 3 2 e *c52*x - 2*c55*x *y - 2*c55*x h_1=------------------------------------- 2*x*y e 4 c55*x h_2=-------- 2*x*y e Free functions or constants : c52 c55 c66 ************************************************************************** CRACK needed : 8530 ms GC time : 570 ms 2 4 x*y 2 A first integral is: (df(y,x) *c55*x + e *df(y,x)*c52*x 3 2 x*y x*y - 2*df(y,x)*c55*x *y - 2*df(y,x)*c55*x - e *c52*x*y - e *c52 2 2 2*x*y + c55*x *y + 2*c55*x*y + c55)/e and an integrating factor: 2 2 x*y x *(2*df(y,x)*c55*x + e *c52 - 2*c55*x*y - 2*c55) ------------------------------------------------------ 2*x*y e free constants: c52 c55 .......................................................................... An example of the determination of a Lagrangian for an ODE Determination of a Lagrangian L for: 2 df(y,x,2)=x + 6*y 2 The ansatz: L = df(y,x) *u_ + v_ This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: 2 2 0= - d_y(1) *df(u_,y) - 2*d_y(1)*df(u_,x) + df(v_,y) - 2*u_*x - 12*u_*y functions: u_(y,x) v_(y,x) separation w.r.t. d_y(1) new function: c67(y) new function: c68(y) integrated equation : 0=c68 + u_ new function: c69(x) new function: c70(x) integrated equation : 0=c70 + u_ separation yields 3 equations substitution : u_= - c70 u_= - c70 v_(y,x) generalized separation new constant: c71 new constant: c72 separation yields 3 equations substitution : c68= - c71 u_= - c70 v_(y,x) substitution : c70= - c71 u_=c71 v_(y,x) substitution : c71= - c72 u_= - c72 v_(y,x) new function: c73(x) new function: c74(x) integrated equation : 3 0=2*c72*x*y + 4*c72*y + c74 + v_ substitution : 3 v_= - 2*c72*x*y - 4*c72*y - c74 u_= - c72 3 v_= - 2*c72*x*y - 4*c72*y - c74 End of this CRACK run The solution : 3 v_= - 2*c72*x*y - 4*c72*y - c74 u_= - c72 Free functions or constants : c74(x) c72 ************************************************************************** CRACK needed : 240 ms GC time : 0 ms 2 2 The solution: L = - (2*(x + 2*y )*y + df(y,x) ) .......................................................................... An example of the factorization of an ODE Differential factorization of: 2 2 df(y,x) - df(y,x)*f*y - q*y df(y,x,2)=------------------------------- y The ansatz: df(y,x) = a#*y + b# This is CRACK - a solver for overdetermined partial differential equations Version 1995-03-20 ************************************************************************** equations: 2 2 2 2 0=df(a#,x)*y + df(b#,x)*y - a#*b#*y + a#*f*y - b# + b#*f*y + q*y functions: a#(x) b#(x) separation w.r.t. y new constant: c75 integrated equation : int(f,x) int(f,x) 0=e *a# + int(e *q,x) - c75 separation yields 3 equations substitution : b#=0 a#(x) b#=0 substitution : int(f,x) - int(e *q,x) + c75 a#=----------------------------- int(f,x) e int(f,x) - int(e *q,x) + c75 a#=----------------------------- int(f,x) e b#=0 End of this CRACK run The solution : b#=0 int(f,x) - int(e *q,x) + c75 a#=----------------------------- int(f,x) e Free functions or constants : c75 ************************************************************************** CRACK needed : 530 ms GC time : 0 ms int(f,x) int(1/e ,x)*c75 e *c76 The solution y=-------------------------------------- int(f,x) int(f,x) int(int(e *q,x)/e ,x) e is the general solution of the original ODE (TIME: crack 23309 24969)